Image Reconstruction for 3D Lung Imaging - Department of Systems ...

a regularization penalty term a much broader class of functions are therefore allowed to be
the solution of the inverse problem, including functions with discontinuities. Another way
to understand the differences with other techniques is to consider the discretized version of
equation 7.6. Suppose that the conductivity is described by piecewise constant elements,
the TV of the 2D image can be expressed as the sum of the TV of each of the k edges, with
each edge weighted by its length:
TV (σ) = � � �
lk
�σm(k) − σ �
n(k)
(7.8)
k
where lk is the length of the k th edge in the mesh, m(k) and n(k) are the indices of the
two elements on opposite sides of the k th edge, and the index k ranges over all the edges.
Equation 7.8 can be expressed in terms of matrices as:
TV (σ) = �
|Lkσ| (7.9)
k
where L is a sparse matrix, with one row per each edge in the mesh. Every row Lk has
two non zero elements in the columns m(k) and n (k) : Lk = [0,...,0,lk,0,...,0, −lk,0...0].
TV regularization is therefore of the ℓ 1 kind: it is a sum of absolute values, in this case a
sum of vector lengths. The absolute value guarantees the positivity of the penalty function
but unfortunately results in non–differentiability in the points where σ m(k) = σ n(k). The
numerical problem thus needs to be addressed properly. However, the important gain is
that the ℓ 1 regularization does not penalize discontinuities.
7.2.3.1 Solving TV - Early Approaches.
Two different approaches were proposed for application of TV to EIT, the first by Dobson
and Santosa [44] and the second by Somersalo et al [109] and Kolehmainen [85]. Dobson
and Santosa replace the absolute value function in the neighbourhood of zero by a polynomial
to obtain continuously differentiable function upon which steepest descent is then
used to perform the minimization. Their approach is suitable for the linearized problem
but suffers from poor numerical efficiency. Somersalo and Kolehmainen successfully applied
Markov Chain Monte Carlo (MCMC) methods to solve the TV regularized inverse problem.
The advantage in applying MCMC methods over deterministic methods is that they do not
suffer from the numerical problems involved with non-differentiability of the TV functional;
they do not require ad hoc techniques. Probabilistic methods, such as MCMC, offer central
estimates and error bars by sampling the posterior probability density of the sought
parameters (therefore differentiability is not required). The sampling process involves a
substantial computational effort, often the inverse problem is linearized in order to speed
up the sampling. What is required is an efficient method for deterministic Tikhonov style
regularization, to calculate a non-linear TV regularized inversion in a short time.
Examination of the literature shows that a variety of deterministic numerical methods
have been used for the regularization of image de-noising and restoration problems with the
TV functional (a good review is offered by Vogel in [117]). The numerical efficiency and
stability are the main issues to be addressed. Use of ad hoc techniques is common, given the
poor performance of traditional algorithms. Most of the deterministic methods draw from
ongoing research in optimization, as TV minimization belongs to the important classes of
problems known as “Minimization of sum of norms” [11][13][39] and “Linear ℓ 1 problems”
[19][125].
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